Independence of volume and genus $g$ bridge numbers

TL;DR

This paper demonstrates that hyperbolic volume and genus g bridge numbers of knots are independent, providing explicit examples of sequences with unbounded and bounded values in each parameter, challenging previous assumptions of correlation.

Contribution

It establishes the independence of hyperbolic volume and genus g bridge numbers for knots, contrasting with known relations in Heegaard splittings.

Findings

Constructed sequences with bounded volume and unbounded bridge number.

Constructed sequences with bounded bridge number and unbounded volume.

Showed that volume and bridge number are completely independent parameters.

Abstract

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a $(g,b)$-bridge surface for a knot $K$ in $S^3$ carries any geometric information related to the knot exterior. In this paper, we show that (unlike in the case of Heegaard splittings) hyperbolic volume and genus $g$ bridge numbers are completely independent. That is, for any $g$, we construct explicit sequences of knots with bounded volume and unbounded genus $g$ bridge number, and explicit sequences of knots with bounded genus $g$ bridge number and unbounded volume.